Non-Hausdorff manifold

In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff. The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line. This is the quotient space of two copies of the real line with the equivalence relation This space has a single point for each nonzero real number and two points and A local base of open neighborhoods of in this space can be thought to consist of sets of the form where is any positive real number. A similar description of a local base of open neighborhoods of is possible. Thus, in this space all neighbourhoods of intersect all neighbourhoods of so the space is non-Hausdorff. The space is however locally Hausdorff in the sense that each point has a Hausdorff neighbourhood. Further, the line with two origins does not have the homotopy type of a CW-complex, or of any Hausdorff space. Similar to the line with two origins is the branching line. This is the quotient space of two copies of the real line with the equivalence relation This space has a single point for each negative real number and two points for every non-negative number: it has a "fork" at zero. The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff. (The etale space is Hausdorff if it is a sheaf of functions with some sort of analytic continuation property.) Because non-Hausdorff manifolds are locally homeomorphic to Euclidean space, they are locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff).